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Stability of motion: method of integral inequalities. Ed. and with a preface by Yu. A. Mitropol’skiĭ. (Устойчивость движения: метод интегральных неравенств.) (Russian) Zbl 0751.70014
Kiev: Naukova Dumka. 272 p. (1989).

This monograph gives a systematic survey of the results related to the development of two classical branches of the theory of differential equations and mechanics, one being the theory of integral inequalities initiated by Gronwall, Kamke and Chaplygin, the other being the theory of stability of motion founded by Lyapunov. The material is divided into five chapters.

Chapter I presents linear integral inequalities, beginning with the Gronwall-Bellman inequality and its generalizations, then inequalities for multiple integrals, functions of several independent variables, discrete systems and interval functions, finally ending with inequalities for abstract operators.

Chapter II describes nonlinear integral inequalities. First it introduces the results obtained by means of direct transformation, Bihari’s method and Lakshmikantham’s method. Then, some other kinds of nonlinear integral inequalities such as discrete systems, two-inequalities systems, interval-valued function systems and operator systems are discussed.

Chapter III gives applications of the inequalities studied in the previous chapters to problems of stability of motion. It is shown that utilization of integral inequalities not only leads to some new stability criteria, but also provides conditions under which the stability of motion of a nonlinear system can be decided by examining the corresponding reduced system.

Chapter IV contributes to the study of stability of motion by use of integral inequalities combined with the Lyapunov functions (scalar, vector and matrix ones). This technique is used to investigate a large number of complicated systems.

Chapter V studies the problem of stability of motion of systems under constant (persistent) perturbation. By means of integral inequalities and Lyapunov’s direct method, the qualitative behaviour of the solutions is estimated, and conditions of stability are obtained. As applications, some mechanical systems under constant perturbation or with slightly variable constraints are considered. A list of 169 references is given.

This is an interesting book both for mathematicians and for researchers engaged in the study of stability of motion in mechanics.

Reviewer: Li Li (Beijing)

MSC:
70K20Stability of nonlinear oscillations (general mechanics)
70-02Research monographs (mechanics of particles and systems)
34DxxStability theory of ODE
34-02Research monographs (ordinary differential equations)